I think there are a few things here. Really, the prescriber should be able to assess your Suitability.
Secondly, if your NHS GP is giving you advice about this, then it becomes a nit of a grey area of, well you think I can/ should/ it would be beneficial to my health, why aren't your prescribing it?
Also, depending on how long your GP has been practising, they may know very little about the jab itself.
When Saxenda was V V new, I went to my GP about my side affects he said, "I'm sorry, but I don't really have any answers." I did a bit of a Google and said, eat more!
Now I'm on mounjaro, I'm not sure how, but my injection site is infected, my GP wasn't inclined to treat me, it seems.
Lastly, can I just recommend shemed for mounjaro? It's about £150 a month, whatever dosage you have, and they can monitor your progress with your HBA1C, cholesterol, and many other things, plus you get a weekly check-in with them.